As in Remark REF  \((A,u)\)  is a minimizer of \(\widetilde{\mathcal {F}}^{\lambda }\)  if and only if
\((A,u+u_0)\)  minimizes the \( \widehat{\widetilde{\mathcal {F}}^{\lambda }}(\cdot ):=\widetilde{\mathcal {F}}^{\lambda }(\cdot -u_0).\)  Thus, we can introduce the following localized version of \(\widetilde{\mathcal {F}}\)  in open subsets \(O\)  of \(\Omega \)  which does not see the substrate:
\(\widetilde{\mathcal {F}}(B,v;O):=\widetilde{\mathcal {S}}(B;O) + \mathcal {W}(B,v; O)\) 
where
\(\widetilde{\mathcal {S}}(B,v;O):= \int _{O\cap \partial ^*B} \varphi (y,\nu _B)d\mathcal {H}^1+ 2\int _{O\cap B^{(1)}\cap \partial B\cap S_v} \varphi (y,\nu _B)d\mathcal {H}^1,\) 
the \(\mathcal {W}(\cdot ;O)\)  is given as in (REF )
and \(S_v^A\)  is defined as in (REF ).
Then the minimality of \((A,u)\)  implies that \((A,u+u_0)\)  is a quasi-minimizer of \(\widetilde{\mathcal {F}}(\cdot ;O)\) 
in \(O,\)  namely,
\(\widetilde{\mathcal {F}}(A,u +u_0;O) \le \widetilde{\mathcal {F}}(B,v;O) + \lambda _0|A\Delta B|\) 
whenever \((B,v)\in \widetilde{\mathcal {C}}\)  with \(A\Delta B\subset \subset O\)  and \(\mathrm {supp}\,(u+u_0-v)\subset \subset O.\) 
Now the proof of the existence of \(\varsigma ^*\)  and \(R^*\)  satisfying (REF ) and (REF ) runs along the same lines of the proof of Theorem REF  for \(m=\infty ,\)  therefore, we do not repeat it here. Note that \(\varsigma ^*\)  and \(R^*\)  depend only on \(c_i\)  and \(\lambda .\)
Let \(A_{\circ }:=\mathrm {Int}(A^{(1)}).\)  We claim that
\(\partial A_{\circ } = \overline{\partial ^*A}.\) 
Indeed, note that \(A^{(1)}\setminus A_{\circ }\subset \partial A^{(1)} =\overline{\partial ^*A},\)  where in the equality we used \(\overline{\partial ^*A} = \overline{\partial ^*A^{(1)}}=\partial A^{(1)}\)  see e.g., [1]}. Thus, \(A_{\circ }\)  is also equivalent to \(A,\)  and hence, \(\partial ^*A_{\circ }=\partial ^*A=\partial ^*A^{(1)}.\)  In particular, \(\partial A^{(1)}=\overline{\partial ^*A_{\circ }}\subset \partial A_{\circ }.\)  On the other hand, assume that there exists \(x\in \partial A_{\circ }\setminus \partial A^{(1)}.\)  Since \(\partial A^{(1)}\)  is closed, there exists \(r>0\)  such that \(\overline{B_r(x)}\cap \partial A^{(1)}=\emptyset .\)  Hence, either \(B_r(x)\subset \mathrm {Int}(A^{(1)})=A_{\circ }\)  or \(\overline{B_r(x)}\cap \overline{A^{(1)}}=\emptyset .\)  Since \(A_{\circ }\)  is open and \(x\in \partial A_{\circ },\)  the inclusion \(B_r(x)\subset A_{\circ }\)  is not possible. On the other hand, since \(\overline{A_{\circ }}\subset \overline{A^{(1)}}\)  and \(x\in \partial A_{\circ },\)  the relation \(\overline{B_r(x)}\cap \overline{A^{(1)}}=\emptyset \)  is also not possible. Thus, \(\partial A_{\circ }\subseteq \partial A^{(1)}.\)
To prove (REF ) we fix \(\Omega ^{\prime }\subset \subset \Omega .\) 
We claim that \(\widetilde{u}\big |_{A_{\circ }}\)  is a minimizer of Griffith functional \(\mathcal {G}:GSBD^2({\rm Int}{(A_{\circ }\cup S\cup \Sigma )}) \rightarrow \mathbb {R},\)
\(\mathcal {G}(v):=\int _{A_{\circ }\cap J_{v}} \varphi (x,\nu _{J_v}) d\mathcal {H}^1 + \int _{A_{\circ }} x)e(v):e(v)dx\) 
with Dirichlet boundary condition \(v=\widetilde{u}=u\)  in \(A_{\circ }\setminus \Omega ^{\prime }.\)  Indeed, for every \(v\in GSBD^2(A_{\circ })\)  with \(\widetilde{u}=v\)  in \(A_{\circ }\setminus \Omega ^{\prime }\)  we define \(B:=A_{\circ }\setminus \overline{J_v}.\)  Then \((B,v)\in \widetilde{\mathcal {C}}\)  and by the minimality of \((A,u)\)
\(\mathcal {G}(u) - \mathcal {G}(v) =\widetilde{\mathcal {F}}(A,u) - \widetilde{\mathcal {F}}(B,v)\le 0.\) 
Since \(S_v^B = J_{\widetilde{u}\big |_{A_{\circ }}}\)  up to a \(\mathcal {H}^1\) -negligible set, (REF ) directly follows from the density estimates for the jump set of Griffith minimizers (see e.g. [2]}) with possibly smaller \(\varsigma ^*\in (0,1)\)  and \(R^*>0.\)
Finally, we prove (REF ) only for \(S_u^A,\)  the other being similar. Let \(\Gamma :=\lbrace x\in S_u^A:\, \theta _*(S_u^A,x)>0\rbrace .\)  Note that \(S_u^c=\bar{\Gamma }.\)
We claim that
\(\mathcal {H}^1(A_{\circ }\cap (\bar{\Gamma }\setminus \Gamma ))=0.\) 
Indeed, let \(\mu :=\mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits \Gamma .\)  Then \(\mu (\bar{\Gamma }\setminus \Gamma )=0.\)  By the regularity of \(\mu ,\)  for every \(\epsilon >0\)  there exists an open set \(U\subset \mathbb {R}^2\)  such that \(L:=A_{\circ }\cap (\bar{\Gamma }\setminus \Gamma ) \subset U\)  and \(\mu (U)=\mathcal {H}^1(U \cap \Gamma )<\epsilon .\)  Note that \(\bar{\Gamma }\subset \overline{\lbrace y\in \Omega \cap J_{\widetilde{u}}:\,\theta _*(J_{\widetilde{u}},x)>0\rbrace },\)  where \(\widetilde{u}\)  is given by Theorem REF . Hence, for (REF ) holds for all points of \(\bar{\Gamma }.\)  By the definition of the closure, and Vitali Covering Lemma we can find at most countable pairwise disjoint family \(\lbrace \overline{B_{r_i}(x_i)}\rbrace _i\)  of closed balls \(\overline{B_{r_i}(x_i)}\)  with \(x_i\in A_{\circ }\cap \Gamma ,\)  \(r_i\le \min \lbrace R^*,\epsilon ,\mathrm {dist}(x,\partial \bar{A})\rbrace \)  such that \(A_{\circ }\cap (\bar{\Gamma }\setminus \Gamma ) \subset \cup _i \overline{B_{5r_i}(x_i)}.\)  Without loss of generality we may assume that \(B_{r_i}(x_i)\subset U.\)  Since \(Q_{r_i/\sqrt{2}}(x_i)\subset B_{r_i}(x_i)\subset Q_{r_i}(x_i),\)  from the definition of Hausdorff premeasure, (REF ) and disjointness of \(\lbrace B_{r_i}(x_i)\rbrace \)  as well as the choice of \(U\)  we obtain
\(\begin{aligned}\mathcal {H}_{10\epsilon }(A_{\circ }\cap (\bar{\Gamma }\setminus \Gamma ))\le & \sum \limits _{i\ge 1} 2\pi (5r_i) \le \frac{10\pi \sqrt{2}}{\varsigma ^*} \sum \limits _{i\le 1} \mathcal {H}^1(Q_{r_i/\sqrt{2}}(x_i) \cap \Gamma ) \\=& \frac{10\pi \sqrt{2}}{\varsigma ^*}\, \mathcal {H}^1\big (\cup _i Q_{r_i/\sqrt{2}}(x_i) \cap \Gamma \big ) \le \frac{10\pi \sqrt{2}}{\varsigma ^*}\, \mathcal {H}^1\big (U \cap \Gamma \big ) <\frac{10\pi \sqrt{2}\epsilon }{\varsigma ^*}.\end{aligned}\) 
Now letting \(\epsilon \rightarrow 0\)  we get (REF ).
